4 ms·

I'm no mathematician, but I have always found it strange infinities are talked about as physical states (towers of tall towers, etc), and not functions. Intege

by dexwiz 5y ago

I'm no mathematician, but I have always found it strange infinities are talked about as physical states (towers of tall towers, etc), and not functions. Integer number counting is essentially a successor function; take N, add 1, output N+1. One input, one output. Real number counting is a bit more loose. To find the numbers between .1 and .2, you find all fractionals of a given size, normally 1/10. So we get .10, .11, .12, etc. This function gives more outputs than inputs, thus an increase in cardinality. Forcing just seems like changing the function to increase the cardinality. In my mind, all infinities are equal, it's how you generate them that matters.

IncRnd 5y ago
I'm not a mathematician, either, but I believe your comment shows why not all infinities are equal. When you attempt to enumerate .1, .2, .3, .4, etc. you get an infinite count of numbers. However, when you perform that same operation on .05, .15, .2, .25, and so on, you will get "twice as many numbers" as the first time around. In this way, the second infinite sequence has "twice as many numbers" as the first infinite sequence.
- gowld 5y ago
  No, it doesn't. Then have the same number, as can be plainly seen by lining them up and counting the elements in order.
zeugmasyllepsis 5y ago
> Real number counting is a bit more loose. To find the numbers between .1 and .2, you find all fractionals of a given size, normally 1/10. Unfortunately the fractionals method does not work, since there are numbers (in fact, uncountably infinitely many!) irrational numbers which cannot be expressed as fractions, which are nonetheless real numbers (such as the square root of two and pi).
drdeca 5y ago
There is a way of mapping different ordinal numbers which are less than some large countable ordinal (idr how high up it goes. I suspect it isn’t well defined at least past the church kleene ordinal, but I think it is much before that, but also I could be totally wrong) to different functions from integers to integers which increase at different rates. These methods include the Hardy hierarchy, and the fast growing hierarchy. (But regarding all infinities being equal, that’s just incorrect, sorry.)

by dexwiz 5y ago

IncRnd 5y ago

I'm not a mathematician, either, but I believe your comment shows why not all infinities are equal. When you attempt to enumerate .1, .2, .3, .4, etc. you get an infinite count of numbers. However, when you perform that same operation on .05, .15, .2, .25, and so on, you will get "twice as many numbers" as the first time around. In this way, the second infinite sequence has "twice as many numbers" as the first infinite sequence.

gowld 5y ago
No, it doesn't. Then have the same number, as can be plainly seen by lining them up and counting the elements in order.

zeugmasyllepsis 5y ago

> Real number counting is a bit more loose. To find the numbers between .1 and .2, you find all fractionals of a given size, normally 1/10. Unfortunately the fractionals method does not work, since there are numbers (in fact, uncountably infinitely many!) irrational numbers which cannot be expressed as fractions, which are nonetheless real numbers (such as the square root of two and pi).

drdeca 5y ago

There is a way of mapping different ordinal numbers which are less than some large countable ordinal (idr how high up it goes. I suspect it isn’t well defined at least past the church kleene ordinal, but I think it is much before that, but also I could be totally wrong) to different functions from integers to integers which increase at different rates. These methods include the Hardy hierarchy, and the fast growing hierarchy. (But regarding all infinities being equal, that’s just incorrect, sorry.)